No, that's not how it works. So far as I know, FOIA basically never compels private entities to do anything. It can compel usage by the public body, in many (but not all) cases. But FOIA isn't going to let you crack open an Atlassian (or Flock) database, unless something truly novel happens in FOIA jurisprudence.

We might be talking past each other, because this stuff is subtle. But basically: whoever's doing the actual document production under FOIA, it's got to be a public body. If you're a commercial SAAS serving a public body, and you've got data that FOIA says needs to be produced, that's the public body's problem, not yours.